Example 1. The function f⁢(x,y)=x2+y2+1 from ℝ2 to ℝ has a (global) minimum point (0, 0), where its partial derivatives∂⁡f∂⁡x=2⁢x and
∂⁡f∂⁡y=2⁢y both equal to zero.

Example 2. Also the function g⁢(x,y)=x2+y2 from ℝ2 to ℝ has a (global) minimum in (0, 0), where neither of its partial derivatives ∂⁡g∂⁡x and
∂⁡g∂⁡y exist.

Example 3. The function f⁢(x,y,z)=x2+y2+z2 from ℝ3 to ℝ has an absolute minimum point (0, 0, 0), since ∇⁡f=2⁢x⁢𝐢+2⁢y⁢𝐣+2⁢z⁢𝐤=𝟎⟹x=y=z=0, ∂2⁡f∂⁡x2=∂2⁡f∂⁡y2=∂2⁡f∂⁡z2=2>0, and f⁢(0, 0, 0)≤f⁢(x,y,z) for all (x,y,z)∈ℝ3.